If λ1, λ2 and λ3 are the wavelengths of the waves giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe. Then, the ratio of wavelengths λ1: λ2: λ3 is:

Question

If λ1, λ2 and λ3 are the wavelengths of the waves giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe. Then, the ratio of wavelengths λ1: λ2: λ3 is: (A) 1: 3: 5 (B) 1: 2: 3 (C) 5: 3: 1 (D) 1: (1/3): (1/5)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/8e94e40388346a133eb7368c7cfcf716-.png" /> ℓ=(λ1/2) ⇒ λ1=2 ℓ ℓ=(3 λ2/2) ⇒ λ2=(2 ℓ/3) ℓ=(5 λ3/2) ⇒ λ3=(2 ℓ/5) ∴ λ1: λ2: λ3=2 ℓ: (2 ℓ/3): (2 ℓ/5) =1: (1/3): (1/5)

Q. If $λ_{1}, λ_{2}$ and $λ_{3}$ are the wavelengths of the waves giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe. Then, the ratio of wavelengths $λ_{1} : λ_{2} : λ_{3}$ is:

1 : 3 : 5

1 : 2 : 3

5 : 3 : 1

$1 : \frac{1}{3} : \frac{1}{5}$

$ℓ = \frac{λ _{1}}{2} \Rightarrow λ_{1} = 2 ℓ$
$ℓ = \frac{3 λ _{2}}{2} \Rightarrow λ_{2} = \frac{2 ℓ}{3}$
$ℓ = \frac{5 λ _{3}}{2} \Rightarrow λ_{3} = \frac{2 ℓ}{5}$
$∴ λ_{1} : λ_{2} : λ_{3} = 2 ℓ : \frac{2 ℓ}{3} : \frac{2 ℓ}{5}$
$= 1 : \frac{1}{3} : \frac{1}{5}$

Q. If λ1​,λ2​ and λ3​ are the wavelengths of the waves giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe. Then, the ratio of wavelengths λ1​:λ2​:λ3​ is: